Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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3b(123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.012925023376&{}:{}&1.573369701203&{}:{}&-0.560444677827&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.121334904069&{}:{}&1.006450210230&{}:{}&-0.127785114299&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.209585943868&{}:{}&2.403780614283&{}:{}&-1.194194670415&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.479876041339&{}:{}&1.933295137058&{}:{}&-1.413171178397&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.182661048441&{}:{}&3.122132208906&{}:{}&-1.939471160465&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{3b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3b}}\) Radical center of the Malfatti circles
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&1.278649238600&{}:{}&3.343790863202&{}:{}&-3.622440101802&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.249218138163&{}:{}&1.897185297387&{}:{}&-0.647967159224&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.306997126177&{}:{}&5.536793102824&{}:{}&-2.229795976647&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.449841245026&{}:{}&2.252028762085&{}:{}&-0.802187517059&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.071447962111&{}:{}&4.169518189229&{}:{}&-3.098070227118&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.017388286049&{}:{}&2.116684947350&{}:{}&-1.099296661302&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.177116136499&{}:{}&1.348305274138&{}:{}&-0.171189137639&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.390329829583&{}:{}&1.020750084069&{}:{}&-0.411079913652&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.154755857176&{}:{}&1.283670730899&{}:{}&-0.438426588075&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&3.716989200105&{}:{}&-2.716989200105&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.514173943788&{}:{}&0.000000000000&{}:{}&1.514173943788&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.198857024995&{}:{}&0.801142975005&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-1.671895431601&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.299061765796&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.768396551412&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.278649238600&{}:{}&3.343790863202&{}:{}&-3.622440101802&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.249218138163&{}:{}&1.897185297387&{}:{}&-0.647967159224&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.306997126177&{}:{}&5.536793102824&{}:{}&-2.229795976647&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.012925023376&{}:{}&1.573369701203&{}:{}&-0.560444677827&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.449841245026&{}:{}&2.252028762085&{}:{}&-0.802187517059&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.071447962111&{}:{}&4.169518189229&{}:{}&-3.098070227118&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.017388286049&{}:{}&2.116684947350&{}:{}&-1.099296661302&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.121334904069&{}:{}&1.006450210230&{}:{}&-0.127785114299&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.177116136499&{}:{}&1.348305274138&{}:{}&-0.171189137639&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.390329829583&{}:{}&1.020750084069&{}:{}&-0.411079913652&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.154755857176&{}:{}&1.283670730899&{}:{}&-0.438426588075&. \end{alignedat} \]
3b (123)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.209585943868&{}:{}&2.403780614283&{}:{}&-1.194194670415&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.278649238600&{}:{}&3.343790863202&{}:{}&-3.622440101802&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.249218138163&{}:{}&1.897185297387&{}:{}&-0.647967159224&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.306997126177&{}:{}&5.536793102824&{}:{}&-2.229795976647&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.449841245026&{}:{}&2.252028762085&{}:{}&-0.802187517059&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.071447962111&{}:{}&4.169518189229&{}:{}&-3.098070227118&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.017388286049&{}:{}&2.116684947350&{}:{}&-1.099296661302&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.479876041339&{}:{}&1.933295137058&{}:{}&-1.413171178397&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&3.716989200105&{}:{}&-2.716989200105&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.514173943788&{}:{}&0.000000000000&{}:{}&1.514173943788&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.198857024995&{}:{}&0.801142975005&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.182661048441&{}:{}&3.122132208906&{}:{}&-1.939471160465&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.449841245026&{}:{}&2.252028762085&{}:{}&-0.802187517059&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.071447962111&{}:{}&4.169518189229&{}:{}&-3.098070227118&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.017388286049&{}:{}&2.116684947350&{}:{}&-1.099296661302&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&3.716989200105&{}:{}&-2.716989200105&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.514173943788&{}:{}&0.000000000000&{}:{}&1.514173943788&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.198857024995&{}:{}&0.801142975005&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)